Problem: Simplify the following expression: $ a = \dfrac{4}{7} + \dfrac{5t + 2}{-9t - 5} $
Answer: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-9t - 5}{-9t - 5}$ $ \dfrac{4}{7} \times \dfrac{-9t - 5}{-9t - 5} = \dfrac{-36t - 20}{-63t - 35} $ Multiply the second expression by $\dfrac{7}{7}$ $ \dfrac{5t + 2}{-9t - 5} \times \dfrac{7}{7} = \dfrac{35t + 14}{-63t - 35} $ Therefore $ a = \dfrac{-36t - 20}{-63t - 35} + \dfrac{35t + 14}{-63t - 35} $ Now the expressions have the same denominator we can simply add the numerators: $a = \dfrac{-36t - 20 + 35t + 14}{-63t - 35} $ $a = \dfrac{-t - 6}{-63t - 35}$ Simplify the expression by dividing the numerator and denominator by -1: $a = \dfrac{t + 6}{63t + 35}$